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The (3,4,∞)(3,4,\infty) Modular Family of 2-Tori as a Complex Structure on S6S^6

Hopf's problem, posed in 1948: does the six-sphere S6S^6 admit an integrable complex structure? S6S^6 is one of only two spheres carrying an almost complex structure at all (the other is S2S^2), from the octonions' multiplication, but almost complex structures need not be integrable, and whether that one - or any other - integrates has stood open for 78 years through a history of disputed attempts, including a widely discussed 2016 argument by Atiyah that did not hold up. This paper claims yes: it builds an explicit compact complex threefold XX, fibred over P1\mathbb{P}^1 by complex 2-tori degenerating at three points, and argues XX is simply connected with the integral homology of S6S^6, hence diffeomorphic to it.

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Construction
Field
Complex geometry; differential topology
Posed by
Heinz Hopf
Year posed
1948
Years open
78y
Solved
2026-08-24
Model
Claude
Vendor
Anthropic
Collaborators
Levent Alpöge
Verification
Unreviewed
Publication
Preprint
Significance
65 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims an explicit compact complex threefold XX, fibred over P1\mathbb{P}^1 by complex 2-tori via period functions on the (3,4,∞)(3,4,\infty) orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues XX is simply connected with H∗(X;Z)=H∗(S6;Z)H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z}), hence diffeomorphic to S6S^6, with algebraic dimension exactly 1. This directly contradicts [CDP20, Cor. 2.3], a published (and once-corrected) theorem; the paper states this and argues where the two accounts diverge, rather than overlooking it. Posted hours before this entry, with no independent check, no formalisation, and no refutation yet in any venue found. Filed as a candidate specifically because none of that has happened, not because a problem with the argument has been found.

What the AI did

The manuscript itself, 108 pages read in full here, contains no AI disclosure of any kind: no acknowledgments section, no methods note, and not one occurrence of "Claude", "Anthropic", "AI" or a paraphrase anywhere in its text. The only disclosure is the author's own public post, reproduced by third-party tech coverage since no stable direct link to it could be confirmed: "Please welcome to the world a beautiful new geometric object... claude really contains multitudes :D Does S^6 admit a complex structure? Yup." That names the model and credits it substantively but says nothing about which parts of a 108-page argument it produced, checked, or merely discussed - the same shape of vague, first-party, off-paper disclosure this catalog already has from this author on the elliptic-curve rank entries, classified the same way there.

Verification

Read here in full via pdftotext on 24 August 2026, hours after it was posted: a genuine 108-page manuscript with an abstract, eight numbered sections, two appendices and a 60-item bibliography of real, checkable citations (Kodaira, Mumford, Orlik, Hopf's original 1948 paper, and the Campana-Demailly-Peternell papers it contradicts). It states its conflict with [CDP20, Cor. 2.3] explicitly rather than ignoring it, and argues a specific point of divergence (that R2f∗(TX⊗L)=0R^2f_*(T_X\otimes L)=0 for every line bundle LL, tied to the non-normality of one singular fibre) - a paper aware of what it is claiming, which is not evidence that the claim holds. No refutation or independent confirmation has surfaced in the venues checked (a Hacker News thread, exploratory and non-technical so far; a Chinese-language math Q&A). No Lean formalisation and no computational certificate accompanies it, so unlike every other entry this queue has handled, there is no kernel check or exact-arithmetic recomputation available to perform. The mathematics itself - monodromy of the (3,4,∞)(3,4,\infty) triangle group, Kodaira logarithmic transforms, a Mumford-style toric degeneration, a Seifert-fibred homology computation - was not and could not be independently verified here; this classification reflects that fact, not a judgement on the argument's quality.

Sources

Changelog11 changes
  • SilentDingo771commented
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  • LucidManta102changed More links from “problem-record: Agricola, Bazzoni, Goertsches, Konstantis, Rollenske - On the history of t…” to “problem-record: Agricola, Bazzoni, Goertsches, Konstantis, Rollenske - On the history of t…”
  • MellowIbis537commented
  • MellowLemur682commented
  • JollyVulture614commented
  • SilentDingo771commented
  • elicommented
  • MellowLemur682commented
  • Rasmus Lindahladded this entry

Discussion8

SilentDingo77102 Sep 2026, 08:52 UTC

I have my doubts on this 2022 article, since it seems to assume that the set of complex structures on S^6 is path connected which is not obvious. Also it now seems to me that there is indeed an error in CDP, where they reduce the calculation of the cohomology group of a sheaf to that of it's quotient by the torsion which seems not quite accurate. still looking forward to digest of all of this....

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MellowLemur68225 Aug 2026, 21:36 UTC· edited

JollyVulture614 the star vanishes, it is just Serre duality H^n(...)* = H^0(...). Also a vector space V is trivial if and only if its dual V* is trivial. It would be surprising if AI made a simple linear algebra mistake.

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JollyVulture61425 Aug 2026, 17:12 UTC

I was interested in how this contradicted the CDP paper. On page 88 I see a statement about H^2(...)*~=H^0(...)

On the next line down I see ... 'hence H^2(...)!=0'

I'm not an expert, but where did the star go?

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SilentDingo77125 Aug 2026, 13:36 UTC

Just to be cautious there are still some huge uncertenties about this proof since it contradicts a major paper of Campana Demailly Peternell. There has been a lot of proofs about the complex structure on S^6 which turned out to be false. Indeed the signficance is much higher than 65/100 which begs much caution.

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eli24 Aug 2026, 15:01 UTC· edited

This is more significant than the 65/100 lets on; it's pretty comfortably the largest discovery in mathematics using AI.

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MellowLemur68224 Aug 2026, 14:29 UTC

This should be ranked higher then 65. I would say it is easily a 70-75 score problem. It resolves the question of complex structures on spheres completely. One of the most important, famous and long standing questions in complex geometry. The new ratio on Riemann Zeta zeroes on the critical line is just an improvement of a benchmark, it does not resolve a concrete problem. Moreover even proving 100 percent is not the way to prove RH. Therefore I argue that this result should be placed higher then 68. At least 70. Also if the paper turns out to be true, it would be the most impressive AI contribution so far: 100 pages of really advanced, postgraduate level mathematics, multiple technical arguments. The length of the result speaks for the diffuculty of the problem.

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